Re: bijection of R: R <--> Rx.....xR
- From: Martin.Sleziak@xxxxxxxxx
- Date: 7 Sep 2005 11:03:24 -0700
Maybe interesting note about history of this problem:
This map appeared in Cantor's letters to Dedekind. Cantor wanted to
find out whether it is possible to find one-to-one map between the
interval I=[0,1] and the square IxI.
In a letter to Dedekind dated 5 January 1874 he wrote:
"Can a surface (say a square that includes the boundary) be uniquely
referred to a line (say a straight line segment that includes the end
points) so that for every point on the surface there is a corresponding
point of the line and, conversely, for every point of the line there is
a corresponding point of the surface? I think that answering this
question would be no easy job, despite the fact that the answer seems
so clearly to be "no" that proof appears almost unnecessary."
Dedekind's answer was, that it's not possible, because obviously two
independent variables cannot be transformed into one.
Later Cantor sent the above mentioned map as the answer. After Dedekind
noticed a flaw in the proof, Cantor constructed a new - but not so
simple - bijection between IxI and I.
(This was a translation from the Czech book: Balcar, Stepanek: Teorie
mnozin [Set Theory]. Sorry if my English wasn't good enough.)
History seems to repeat - now Peter posted a map and David noted that
it needs a correction.
Martin
================================
Martin Sleziak
Homepage: http://thales.doa.fmph.uniba.sk/sleziak/,
http://msleziak.webzdarma.cz
.
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